CUBO A Mathematical Journal Vol.12, N°03, (83–97). October 2010

 

The Semigroup and the Inverse of the Laplacian on the Heisenberg Group¹

 

APARAJITA DASGUPTA AND M.W. WONG

Department of Mathematics, Indian Institute of Science, Bangalore–560012, India email: adgupta@math.iisc.ernet.in

Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada email: mwwong@mathstat.yorku.ca

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ABSTRACT

By decomposing the Laplacian on the Heisenberg group into a family of parametrized partial differential operators L _t ,t ∈ R \ {0}, and using parametrized Fourier-Wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated by L _t, and the inverse of L _t . Using these formulas and estimates, we obtain Sobolev estimates for the one-parameter semigroup and the inverse of the Laplacian.

Key words and phrases: Heisenberg group, Laplacian, parametrized partial differential operators, Hermite functions, Fourier-Wigner transforms, heat equation, one parameter semigroup, inverse of Laplacian, Sobolev spaces.

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RESUMEN

Mediante descomposición del Laplaceano sobre el grupo de Heisenberg en una familia de operadores diferenciales parciales parametrizados L _t, t ∈ R \{0}, y usando transformada de Fourier-Wigner parametrizada, damos fórmulas y estimativas para la continuidad fuerte del semigrupo generado por L _t, y la inversa de L _t. Usando esas fórmulas y estimativas obtenemos estimativas de Sobolev para el semigrupo a un parámetro y la inversa del Laplaceano.

Math. Subj. Class.: 47F05, 47G30, 35J70.

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Notas

¹This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

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